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Theorem iscmgmALT 42725
Description: The predicate "is a commutative magma." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ismgmALT.b 𝐵 = (Base‘𝑀)
ismgmALT.o = (+g𝑀)
Assertion
Ref Expression
iscmgmALT (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ comLaw 𝐵))

Proof of Theorem iscmgmALT
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6437 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 fveq2 6437 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
31, 2breq12d 4888 . . 3 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ (+g𝑀) comLaw (Base‘𝑀)))
4 ismgmALT.o . . . 4 = (+g𝑀)
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5breq12i 4884 . . 3 ( comLaw 𝐵 ↔ (+g𝑀) comLaw (Base‘𝑀))
73, 6syl6bbr 281 . 2 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ comLaw 𝐵))
8 df-cmgm2 42721 . 2 CMgmALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) comLaw (Base‘𝑚)}
97, 8elrab2 3589 1 (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ comLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1656  wcel 2164   class class class wbr 4875  cfv 6127  Basecbs 16229  +gcplusg 16312   comLaw ccomlaw 42686  MgmALTcmgm2 42716  CMgmALTccmgm2 42717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-cmgm2 42721
This theorem is referenced by: (None)
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